The effective initial permeability of a ferrite rod decreases with increasing diameter. This is because the flux is increasingly concentrated toward the outside of the radius. As you go deeper toward the center, the presence of ferrite has a diminishing contribution to the flux. Therefore, for large enough diameters, you can use a hollow cylinder of ferrite with little effect on the inductance as compared to solid ferrite. In other words, a ferrite sleeve. I *think* diameter’s diminishing contribution to total flux begins to become significant with radiuses between 8 and 16 mm.
The effective permeability of a 9 inch or 230 mm diameter slug of mix 61 (u’ = 125) is about 4 or 5. In this case, most of the ferrite isn’t actually being used if it is a (heavy) solid slug. The calculations that predict inductance still work fairly accurately for hollow ferrite cylinders of 8 to 16 mm wall thickness.
The sensitivity of an air-core loop antenna increases with increasing diameter. This also applies if there is ferrite present.
For a fixed length of ferrite, as diameter increases, ferrite’s contribution to sensitivity decreases in proportion to that of an air core loop. (Ferrite permeability is still higher than air, but diminishing) The effective permeability of the ferrite decreases with increasing diameter. At some point in increasing diameter, the presence of ferrite will have a nil effect on the antenna sensitivity. It becomes an air core loop slightly enhanced by the presence of ferrite. I *think* that diameter is somewhere between 1/2 and 1 meter, (beyond which it behaves mostly as an air core loop). My intuition says that diameter may be smaller for shorter lengths of ferrite but I have not confirmed this experimentally.
The converse of this relationship is that increasing length of the ferrite increases the magnetic flux. At longer lengths, the effective permeability begins to approach a limit set by the initial permeability of the ferrite material. Therefore, the benefit gained by further increasing the length diminishes as the length increases.
In between these two extremes there is a practical region where doubling the length has about the same effect on antenna sensitivity as doubling the diameter.
For solid ferrite, going for length is far more efficient because increasing the diameter results in an exponential increase of mass.
For a hollow sleeve consisting of bars or rods mounted on a former, arranged as a cylinder, doubling the length vs. doubling the diameter use the same amount of ferrite. Antenna performance has about a 1 to 1 ratio between proportional changes in diameter vs. proportional changes in length. Doubling the diameter has about the same effect as doubling the length of ferrite.
However, there is one possible advantage of going for length over diameter. Longer ferrite has greater effective permeability. Therefore one may need less Litz wire to attain the same value of inductance. Less wire equals less resistance equals better selectivity and gain. This improvement may be unmeasurable if losses in ferrite are dominant. On the other hand, Litz wire is expensive, so there may still be a cost advantage. It may be also easier to construct a longer ferrite sleeve than a fatter one.
I made my first “ferrite sleeve antenna” on the job in 2005 using 24 mix 61 ferrite rods 6 inches long. These rods originally were vendor samples that were candidates for cost reducing a radio. I built the antenna for the purpose of evaluating another radio prototype, and later, experimenting to see if it was possible to improve reception of MW HD radio. I was using an ibiquity evaluation board. The idea was to overcouple to a tuned resonant antenna to create a two peaked response curve enhancing the two outermost iboc sidebands. The experiment worked beautifully but did not improve HD reception. Then with critical coupling it became a fantastic dx antenna for otherwise mediocre radios. This antenna eventually was donated to a middle school science program.